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Mirror symmetry rupture in double photoionization ofendohedrally on�ned atomsF. D. Colave hia1, G. Gasaneo2, and D. Mitnik3

1Div. Colisiones Atómi as, Centro Atómi o Barilo he and Coni et,8400 S. C. de Barilo he, Río Negro, Argentina2Departamento de Físi a, Universidad Na ional delSur and Coni et, 8000 Bahía Blan a, Argentina and

3Instituto de Astronomía y Físi a del Espa io,Dep. de Físi a, Universidad de Buenos Aires,and Coni et, 1428 Buenos Aires, Argentina(Dated: May 19, 2009)Abstra tWe study the double ele troni emission by photon impa t from He in the enter of a spheri alfullerene, whi h is modeled by a square-well shell. This system exhibits a manifold of avoided rossings as a fun tion of the well depth, and present mirror olapses. However, this symmetry isbroken in the triple di�erential ross se tion due to the delo alization of the He ele trons in theinitial state. Moreover, the fullerene potential involves higher angular momenta partial waves to bein luded in the pro ess, whi h modi�es the well-known two-lobe ross se tion from isolated He.PACS numbers: 31.15.vj, 32.80.Fb, 37.30.+i

1

Fullerene mole ules, formed by pentagonal and hexagonal arrangements of atoms, have re- eived attention right from its dis overy and undoubtely opened many new aptivating areasin physi s. These mole ules ome in di�erent sizes and shapes, from the well known, quasispheri al C60 arbon fullerene [1℄ to nanotubes. Beyond the spe i� features of fullerenes,many methods to in lude atoms inside their shells have been developed [2, 3℄. These om-pounds form stable mole ules that also attra ted the interest of the ommunity, su h as theirpotential use as nano ages to storage atoms [4, 5℄ . Many properties of these systems aredetermined by the di�eren es in the physi s of the embedded atom ompared to the isolatedone.Some of the most interesting of these features arise when these endohedrally embeddedatoms intera t with light and ele troni emission o urs during the pro ess. Sin e the pi-oneering work of Puska and Nieminen [6℄, di�erent kinds of resonant behavior have beenidenti�ed. First eviden e of these e�e ts were observed in the photoionization of C60, re-vealed as os illations in the ross se tions [7℄ and attributed to the ability of the arbonshell to support an intramole ular standing wave. Con�nement resonan es are predi tedfor endohedrally embedded atoms su h as Xe�C60 [8, 9℄, due to the re�e tion of the pho-toele tron in the fullerene age. Furthermore, the role of multiele troni orrelation hasbeen investigated re ently, giving rise to the so- alled orrelation on�nement resonan es[10℄, and also the interferen e among resonan es [11℄. However, all these works involved a omplex multiele troni atom inside the fullerene shell, that gives rise to many me hanismsof ele troni emissions even when the atom is isolated.Few works are devoted to multiple photoionization of fullerenes or endohedrally embed-ded atoms. Kidun et al. have examined the multiple photoionization of C60 within a manyparti le approa h [12℄. Two-ele tron photoionization has been studied for high photon ener-gies by Amusia [13℄, but only the double-to-single photoionization ross se tions ratio werereported. They analized the pro ess with highly asymmetri al energy sharing, where theenergy of one ele tron is mu h bigger that the energy of the other one. The emission inthat ase pro eeds with a shake o� me hanism, where the fast ele tron absorbs most of theenergy of the photon, while the slow one is eje ted due to the residual modi�ed atomi �eldleft after the eje tion of the �rst one. Sin e their model of the fullerene potential was a deltafun tion, it did not take into a ount a urately the possible delo alization of the atomi ele trons into the age. 2

Whereas the investigation of bound or ontinuum states of two ele tron systems anbe performed by di�erent te hniques, double photoionization (DPI) of atomi or mole ularspe ies by single photons has unique advantages. First, it enables to dire tly probe thedynami s of the ele tron pair both in the initial and �nal states with the same ollisionalpro ess. Besides, it is free of long-range orrelations between the target and the in omingparti les. Finally, the omplete absorption of the photon by both ele trons is determinedby the interele troni orrelation. Then, it is not surprising that this pro ess has beenthoroughly studied along the years for both atomi and mole ular spe ies [14℄.In this work we bring a di�erent perspe tive to the problem of DPI from endohedri allyembedded atoms. We hoose the simplest system to study the role of the fullerene age in theele troni emission, as well as the in�uen e of the interele troni orrelation in the pro ess,both in the bound state as well as in the ontinuum one. To this end, we onsider a He atomin the enter of the fullerene age and assume that there are only two a tive ele trons in thesystem. We also fo us on equal energy sharing onditions for the eje ted photoele trons.The simpli�ed ele troni stru ture of the fullerene mole ule seen by an ele tron has beenusually des ribed through a age model potential [6℄:Vw(r) =

−U0 if rc ≤ r ≤ rc + ∆

0 otherwise (1)For C60, rc = 5.75 a.u., ∆ = 1.89 a.u. [6, 7℄. These simple assumptions enable oneto dis over all the ri hness of these systems. Moreover, di�erent spheri al fullerenes aredes ribed varying the well depth. In fa t, the energeti stru ture of these mole ules as afun tion of the magnitude of the age potential U0 presents a manifold of avoided rossingbetween states, even within this two-ele tron model [15, 16℄. Similar results were foundfor endohedrally embedded hydrogen [17℄. For the sake of simpli ity, we hoose to analyzethe rossing between the ground and �rst ex ited states. Thus, the initial state of thesystem will be written in terms of one ele tron 1s (φ1s(r) = 4√

2 exp (−2r)) and 2s (φ2s(r) =

2 exp (−r)(1 − r)) states of the isolated atom, and the approximate solution of the modelpotential (1), φw(r) = exp (−α(r − r0)2) where r0 = rc + ∆/2. We onstru t the following on�guration intera tion (CI) wave fun tion with these one-ele tron wave fun tions for the

3

bound state:Ψi(r1, r2) = N {a ϕ1s(r1)ϕ1s(r2)ϕcorr(r12)

+ b [ϕ1s(r1)φw(r2) + 1 ↔ 2]

+ c [ϕ1s(r1)ϕ2s(r2) + 1 ↔ 2]ϕcorr(r12)}

(2)where ϕcorr(r12) = 1 + r12/2 is a orrelation fa tor.This state is suitable to analyze the ollision pro ess near the rossing between the groundand the �rst ex ited state of the system as a fun tion of the potential depth. The basisparameters a, b, c as well as the exponential fa tor α, the normalization onstant N andthe energy depend on U0 and are obtained with variational te hniques, see Table 1. The rossing is at U cross0 = 1.35 a.u.[18℄ We also hoose two other values of the magnitude ofthe age potential U0 to al ulate the ross se tions, one below (U0 = 1.1 a.u.) and oneabove (U0 = 1.65 a.u.) the rossing of the levels. This parti ular sele tion of the basis doesenhan e the interplay between the free atomi ground state and the partially delo alizedstate with one ele tron in the atom and the other one in the fullerene, sin e the ontributionof the 1s2s state through oe� ient c is mu h smaller than the other ones. Thus, the basisparameters measure the degree of lo alization of the ele trons: when a ≈ 1, both ele tronsare near the atomi ore, while b ≈ 1 implies that the ele troni density spreads up to thefullerene age.We assume a dipolar approximation for the DPI be ause usually experimental data is ob-tained in ollisions with low energy photons [14℄. The triply di�erential ross se tion (TDCS)in terms of the momenta k1 and k2 of the two eje ted ele trons an be obtained in the velo itygauge in terms of the transition matrix T (V )(k1,k2) =

⟨

Ψ−

f (r1, r2) |ε · (∇1 + ∇2)|Ψi (r1, r2)⟩where ε is polarization of the in oming light. We re all that this al ulation an be per-formed also in a eleration or length gauges. Ea h gauge form emphasizes di�erent regionsof the on�guration spa e, but all of them should give the same theoreti al des ription of thepro ess, provided that both the initial Ψi (r1, r2) and �nal Ψ−

f (r1, r2) states are exa t wavefun tions, or at least very good approximations for them. Otherwise, di�eren es betweengauges are apparent [19, 20℄. For simple systems su h as He, these di�eren es are restri tedto the magnitude of the ross se tions, and minor deviations in the angular distributions[21℄ that are irrelevant for the mainly qualitative study presented here. The al ulation ofthe transition matrix is performed by dire t six-dimensional numeri al integration in the4

Table I: Energies and wave fun tions parameters (in atomi units) for ground and �rst ex itedstates of He embedded in the model fullerene age.U0 E N(×10−2) α a b cGround State1.10 -2.8892 4.9390 0.47419 -0.99835 0.00187 0.057231.35 -2.9002 1.54985 0.54823 -0.94505 -0.31958 0.068791.60 -3.0905 0.54473 0.61216 -0.06843 -0.99762 -0.00798First Ex ited State1.10 -2.7127 0.55145 0.47419 0.38490 -0.91623 0.111171.35 -2.8879 4.71427 0.54823 0.99770 -0.03930 -0.055091.60 -2.8890 4.94203 0.61216 0.99837 -0.00609 -0.05671ele troni spheri al oordinates r1 and r2, using a non-deterministi Vegas algorithm, witha relative error is smaller than 3% in the TDCS for all energies and angles onsidered[22℄.Let us turn our attention to the �nal state of the ionized ele trons in the ontinuum.Among many approximate wave fun tions for this state, we hoose a simple C3-style wavefun tion (See [23℄ and referen es therein):

ΨcageC3 (r1, r2) = ψ−

k1(r1)ψ

−

k2(r2)D (α12,k12, r12) (3)where the ele troni wave fun tion in the ombined �eld of the atomi ore (−2/r) and thefullerene age (Vw, eq.(1)) is des ribed by the two-body fun tions ψ−

ki(ri) expanded in partialwaves. For omparison purposes, we also make use of a pure Coulomb wave ψCoul

C3 (r1, r2),where the role of the age potential is negle ted. The ele tron-ele tron orrelation is modeledwith the usual Coulomb distortion fa tor D (α12,k12, r12) = 1F1 (α12, 1, ik12r12 + ik12r12)in terms of the Sommerfeld parameter α12 = 1/k12 and the relative momentum k12 =

k1−k2. This model orre tly reprodu es the asymptoti ondition of the problem. We re allthat, although this model would not lead to a urate absolute di�erential ross se tions; ita ounts for all the features of the ollisional pro ess for this system.We omputed emission in an equal energy sharing situation (E1 = E2 = 10 eV), withlinearly polarized light. The al ulations of TDCS as a fun tion of the magnitude of thefullerene age U0 are displayed in Fig. 1. Overall, the ross se tions inter hange their5

hara ter as the states go through the avoided rossing, due to the mirror ollapse of theinitial state[16℄. This is more evident for �nal Coulomb states ( ompare red lines in Fig. 1awith Fig. 1f; or Fig. 1d with Fig. 1 ).The emission from the ground state below the rossing (Fig. 1d) fully agrees with theisolated atom DPI from a 1s2 state as expe ted, and is similar for both �nal stated used,ψCoul

C3 (r1, r2) or ψcageC3 (r1, r2). Near and beyond the avoided rossing, the presen e of thefullerene age ountera ts the interele troni repulsion, moving the lobes towards the emis-sion dire tion of ele tron 1 (Fig. 1d to Fig. 1f). This is slightly more important in the al ulation that in ludes the age in the �nal state (bla k urves in Fig. 1 online).However, the ex ited state exhibits a more dramati hange along the rossing: the rossse tion hanges from the typi al two-lobe on�guration to a four-lobe one. Thus, the mirror ollapse observed in the intial state wave fun tions is broken in the ross se tions. Thise�e t an be learly seen in Figs. 1a- when ompared with 1d-f, and it is present for both�nal states used, although it is noti eable at this energy only with the non-pure Coulomb�nal state ψcage

C3 (r1, r2) at this emission energy. Both the initial and �nal states ontribute tothis feature. On one hand, the presen e of a delo alized ele tron far from the nu leus in theinitial state slightly hanges the interele troni repulsion. This has already been observedin al ulations of double photoionization from He(1s2s) states [24℄, and were also attributedto the in reasing extension of the initial state. On the other hand, the introdu tion of the age potential in the �nal state ψcageC3 (r1, r2) plays a very important role, restraining theinterele troni repulsion (Fig. 1b) or enhan ing it (Fig. 1 ) for di�erent well depths.It is lear that the emission from the ex ited state deserves further investigation. Thesimple CI that we adopted to des ribe the initial state enables us to isolate the role of ea hCI state in the ross se tions. We omputed the ontribution of ea h CI state separately forthe DPI of the ex ited state, see Fig. 2. In this ase, the energy sharing is E1 = E2 = 50eV, and we hoose a pure Coulomb wave ψCoul

C3 (r1, r2) for the �nal state. Contribution of theCI atomi states results in the typi al emission of ele tron 2 perpendi ular to the dire tionof ele tron 1. However, the ontribution of the atom-well CI state presents a ompletedi�erent stru ture, with three learly distiguishable lobes. It is lear from this �gure thatthe shape of the ross se tion is mainly di tated by the interplay between CI states, andnot by the simple superposition of them. This is learly seen omparing Fig. 2a and 2 ,where the ontribution of the oherent sum determines the ross se tions, while the role of6

the interferen e among states learly de�nes the behavior at the rossing, see Fig. 2b.We an now ompare the behavior of TDCS for di�erent emission energies. While inFig. 1b (E1 = E2 = 10 eV) two lobes are well de�ned when a pure Coulomb �nal waveis used, six lobes an be observed in Fig2b for E1 = E2 = 50 eV. This is not surprising,sin e the ele troni density of the atom-well CI state has a maximum entered at the age.Besides, partial waves with higher single ele tron angular momentum ontribute to the rossse tion for in reasing ele tron energies and are in luded in the pro ess, whi h results in thesestru tures in the ross se tions. This is a unique e�e t due to the presen e of the age,be ause for isolated atoms, the entrifugal barrier inhibits the penetration of high angularmomenta partial waves into the region where the atomi ele troni density is signi� ant. Wehave he ked that this e�e t is enhan ed when the �nal state ΨcageC3 (r1, r2) that in ludes thefullerene age is used.In summary, we have omputed double photoionization ross se tions from He endohedri- ally embedded in spheri al fullerenes. This simple system shows striking di�eren es withthe ele troni emission from isolated atoms. The presen e of the fullerene age determinesthe stru ture of the ross se tions, breaking the initial state mirror ollapse, and enhan ingthe role of higher angular momentum waves of the eje ted ele trons in the pro ess. Thise�e t is more remarkable for higher energies, although it an be seen for small ones, provid-ing that the �nal state also in ludes the age potential. The relation between the presentresults and other os illatory properties found in this pro ess deserves further explorations.F. D. C. would like to a knowledge the �nan ial support of ANPCyT (PICT 04/20548)and Coni et (PICT 5595). G. G. would like to a knowledge the support by PICTR 03/0437of the ANPCYT (Argentina) and PGI 24/F038 of the Universidad Na ional del Sur (Ar-gentina). D. M. a knowledges the support by UBACyT X471, of the Universidad de BuenosAires (Argentina).

[1℄ H. W. Kroto, J. R. Heath, S. C. O'Brien, R. F. Curl, and R. E. Smalley, Nature 318, 162(1985).[2℄ D. S. Bethune, R. D. Johnson, J. R. Salem, M. S. de Vries, and C. S. Yannoni, Nature 366,123 (1993). 7

[3℄ C. S. Yannoni, M. Hoinkis, M. S. de Vries, D. S. Bethune, J. R. Salem, M. S. Crowder, andR. D. Johnson, S ien e 256, 1191 (1992).[4℄ O. V. Pupysheva, A. A. Farajian, and B. I. Yakobson, Nano Letters 8, 767 (2008).[5℄ For a periodi table of endohedrally embedded atoms, seehttp://homepage.ma . om/js hrier/endofullerenes_table.html.[6℄ M. J. Puska and R. M. Nieminen, Phys. Rev. A 47, 1181 (1993).[7℄ Y. B. Xu, M. Q. Tan, and U. Be ker, Phys. Rev. Lett. 76, 3538 (1996).[8℄ M. Y. Amusia, A. S. Baltenkov, and U. Be ker, Phys. Rev. A 62, 012701 (2000).[9℄ M. Y. Amusia, A. S. Baltenkov, V. K. Dolmatov, S. T. Manson, and A. Z. Msezane, Phys.Rev. A 70, 023201 (2004).[10℄ V. K. Dolmatov and S. T. Manson, J. Phys. B 41, 165001 (2008).[11℄ M. Y. Amusia, A. S. Baltenkov, and L. V. Chernysheva, J. Phys. B 41, 165201 (2008).[12℄ O. Kidun, N. Fominykh, and J. Berakdar, Comp. Mat. S ien e 35, 354 (2006).[13℄ M. Y. Amusia, E. Z. Liverts, and V. B. Mandelzweig, Phys. Rev. A 74, 042712 (2006).[14℄ L. Avaldi and A. Huetz, J. Phys. B 38, S861 (2005).[15℄ M. Neek-Amal, G. Tayebirad, and R. Asgari, J. Phys. B 40, 1509 (2007).[16℄ D. M. Mitnik, J. Randazzo, and G. Gasaneo, Phys. Rev. A 78, 062501 (2008).[17℄ J. P. Connerade, V. K. Dolmatov, P. A. Lakshmi, and S. T. Manson, J. Phys. B 32, L239(1999).[18℄ This value for the rossing depends on the model used for the initial state.[19℄ L. U. An arani, G. Gasaneo, F. D. Colave hia, and C. Dal Cappello, Phys. Rev. A 77, 062712(2008).[20℄ A. S. Kheifets and I. Bray, Phys. Rev. A 69, 050701(R) (2004).[21℄ S. P. Lu ey, J. Ras h, C. T. Whelan, and H. R. J. Walters, J. Phys. B 31, 1237 (1998).[22℄ T. Hahn, Comp. Phys. Comm. 168, 78 (2005).[23℄ S. Otranto and C. R. Garibotti, Phys. Rev. A 67, 064701 (2003).[24℄ J. Colgan and M. S. Pindzola, Phys. Rev. A 67, 012711 (2003).8

0.5 1.0 1.5 2.0-4

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ner

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.u)

U0 (a.u.)

uT rS

bC

uT

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a)

e)

b)

f)

c)

Figure 1: (Color online) Three fold di�erential ross se tion (TDCS) for (γ, 2e) ionization of thehelium ground state in a fullerene age, as a fun tion of the angle of one of the eje ted ele tronsθ2. The other one is eje ted at θ1 = 0◦, �xed respe t to the polarization ε. The polarization ve toris set along the x axis, and the impining light towards the page. The ele trons are eje ted withequal energy E1 = E2 = 10 eV. Dotted (orange) and solid (blue) lines represent the ground andex ited state energy as a fun tion of U0, respe tively. TDCS are shown for both ex ited (a, b, )and ground states (d, e, f); and omputed with a pure Coulomb �nal state, dashed (red) line; orwith exa t potential in luding fullerene age, solid (bla k) line. They are shown for U0 = 1.1a.u.(triangles), U0 = 1.35a.u. (squares) and U0 = 1.6 a.u. ( ir les). All TDCS are res aled to one atmaxima.

9

a) b) c)

Figure 2: (Color Online) Contributions of the ea h initial CI states to the TDCS for E1 = E2 =

50eV, al ulated with the pure Coulomb �nal state for the DPI of the ex ited state of endohedri allyembedded He. Kinemati s as well as values of U0 are the same as in Fig. 1a)-1 ). Solid (bla konline) line and squares: full TDCS, solid ( yan online) line, oherent ontribution; dashed (blue)line, TDCS from �rst term, eq. (2); dash-dotted (green) line, TDCS from se ond term, eq (2); anddotted (red) line, TDCS from third term, eq. (2).All TDCS are res aled to one at maxima.

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